On the (3, N) Maurer-Cartan equation

Abstract

Deformations of the 3-differential of 3-differential graded algebras are controlled by the (3, N) Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of N-differential graded algebras, and use these results to study N Lie algebroids.     

Geometry of Maurer-Cartan Elements on Complex Manifolds

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie algebroid theory. In particular, we extend Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology to the realm of extended Poisson manifolds; we establish a sufficient criterion for these to be finite dimensional; we describe how homology and cohomology are related through the Evens-Lu-Weinstein duality module; and we describe a duality on Koszul-Brylinski homology, which generalizes the Serre duality of Dolbeault cohomology.     

Central extensions of quantum current groups

We describe Hopf algebras which are central extensions of quantum current groups. For a special value of the central charge, we describe Casimir elements in these algebras. New types of generators for quantum current algebra and its central extension for quantum simple Lie groups, are obtained. The application of our construction to the elliptic case is also discussed.     

Classification of Simple Linearly Compact n-Lie Superalgebras

We classify simple linearly compact n-Lie superalgebras with n > 2 over a field ${\mathbb{F}}We classify simple linearly compact n-Lie superalgebras with n > 2 over a field \mathbbF{\mathbb{F}} of characteristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive \mathbbZ{\mathbb{Z}}-graded Lie superalgebras of the form L=?_j=-1^n-1 L_j{L=\oplus_{j=-1}^{n-1} L_j}, where dim L _n−1 = 1, L ₋₁ and L _n−1 generate L, and [L _j , L _n−j−1] = 0 for all j, thereby reducing it to the known classification of simple linearly compact Lie superalgebras and their \mathbbZ{\mathbb{Z}}-gradings. The list consists of four examples, one of them being the n + 1-dimensional vector product n-Lie algebra, and the remaining three infinite-dimensional n-Lie algebras.     

Irreducible representations of current algebras in two dimensions

The irreducibility of a class of unitary representations of certain central extensions of current algebras in two dimensions, constructed in a previous paper, is announced. The techniques are extended and supplemented with new elements in order to construct representations of the central extension of Etingof and Frenkel. Irreducibility of the latter is announced.     

Landau-Ginzburg theories for non-abelian quantum Hall states

We construct Landau-Ginzburg effective field theories for fractional quantum Hall states - such as the Pfaffian state - which exhibit non-abelian statistics. These theories rely on a Meissner construction which increases the level of a non-abelian Chem-Simons theory while simultaneously projecting out the unwanted degrees of freedom of a concomitant enveloping abelian theory. We describe this construction in the context of a system of bosons at Landau level filling factor ν = l, where the non-abelian symmetry is a dynamically generated SU(2) continuous extension of the discrete particle-hole symmetry of the lowest Landau level. We show how the physics of quasiparticles and their non-abelian statistics arises in this Landau-Ginzburg theory. We describe its relation to edge theories - where a coset construction plays the role of the Meissner projection — and discuss extensions to other states.     

Dual formulation of classicalW-algebras

By extending the concept of Maurer-Cartan equations, a dual formulation of (classical) nonlinear extensions of the Virasoro algebra is introduced. This dual formulation is closely related to three-dimensional actions which are analogous to a Chern-Simons action. An explicit construction in terms of superfields of theN = 2 superW ₄-algebra is presented.     

Central Charge Extended Supersymmetric Structures for Fundamental Fermions Around Non-Abelian Vortices

Fermionic zero modes around non-abelian vortices are shown that they constitute two N = 2, d = 1 supersymmetric quantum mechanics algebras. These two algebras can be combined under certain circumstances to form a central charge extended N = 4 supersymmetric quantum algebra. We thoroughly discuss the implications of the existence of supersymmetric quantum mechanics algebras, in the quantum Hilbert space of the fermionic zero modes.     

Current algebra and Wess-Zumino model in two dimensions

We investigate quantum field theory in two dimensions invariant with respect to conformal (Virasoro) and non-abelian current (Kac-Moody) algebras. The Wess-Zumino model is related to the special case of the representations of these algebras, the conformal generators being quadratically expressed in terms of currents. The anomalous dimensions of the Wess-Zumino fields are found exactly, and the multipoint correlation functions are shown to satisfy linear differential equations. In particular, Witten's non-abelean bosonisation rules are proven.     

Mirror Extensions of Vertex Operator Algebras

The mirror extensions for vertex operator algebras are studied. Two explicit examples of extensions of affine vertex operator algebras of type A are given which are not simple current extensions.     

S-duality of color-ordered amplitudes

Recently, it has been proposed that the S-matrix elements on the world volume of an abelian D₃-brane are consistent with the Ward identity associated with the S-duality. In this paper we extend this study to the case of multiple D₃-branes. We speculate that the S-matrix elements are consistent with the S-dual Ward identity irrespective of the ordering of the external states. Imposing this symmetry on the particular case of the S-matrix element of one Kalb–Ramond, one transverse scalar and two non-abelian gauge bosons, we will find the linear S-duality transformation of the commutator of two non-abelian gauge field strengths. Using this transformation and the standard S-duality transformations of the supergravity fields, all other non-abelian S-matrix elements of one closed and three open string states can be found by the S-duality proposal. We will show that the predicted S-matrix elements are reproduced exactly by explicit calculations.     

Spectral properties in a class of operators and group representations in nested Hilbert spaces

We characterize the spectrum of the elements of some operator algebras acting in a nested Hilbert space. In one of those algebras we prove the spectral theorem for Hermitian operators and the SNAG theorem for unitary representations of locally compact Abelian groups. Then we extend the SNAG theorem to representations in three more general classes of operators.     

The logarithm of the derivative operator and higher spin algebras ofW ∞ type

We use the notion of the logarithm of the derivative operator to describeW type algebras as central extensions of the algebra of differential operators. We also provide closed formulae for the truncations ofW ₁₊ to higher spin algebras withsM, for allM2. The results are extended to matrix valued differential operators, introducing a logarithmic generalization of the Maurer-Cartan cocycle.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76Sf00098 and in part by the National Science Foundation under grants PHY-85-15857 and PHY-87-17155Address after July 1, 1992: Dept. of Mathematics, Yale University, New Haven, CTO6520, USA     

Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras

We define the Krichever-Novikov-type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. Very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples and KdV-type hierarchies, emphasizing the similarities and differences with the case of smooth symbols on the circle.     

The central extension of Kac-Moody-Malcev algebras

We introduce a class of infinite-dimensional Kac-Moody-Malcev algebras. These algebras are the generalization of Lie algebras of the Kac-Moody type to Malcev algebras. We demonstrate that the central extensions of the Kac-Moody-Malcev algebras are given by the same cocycles as in the case of Lie algebras. Analogues of Kac-Moody-Malcev algebras may be also introduced in the case of an arbitrary Riemann surface.     

Joint Extension of States of Subsystems for a CAR System

 The problem of existence and uniqueness of a state of a joint system with given restrictions to subsystems is studied for a Fermion system, where a novel feature is non-commutativity between algebras of subsystems. For an arbitrary (finite or infinite) number of given subsystems, a product state extension is shown to exist if and only if all states of subsystems except at most one are even (with respect to the Fermion number). If the states of all subsystems are pure, then the same condition is shown to be necessary and sufficient for the existence of any joint extension. If the condition holds, the unique product state extension is the only joint extension. For a pair of subsystems, with one of the given subsystem states pure, a necessary and sufficient condition for the existence of a joint extension and the form of all joint extensions (unique for almost all cases) are given. For a pair of subsystems with non-pure subsystem states, some classes of examples of joint extensions are given where non-uniqueness of joint extensions prevails. Received: 17 May 2002 / Accepted: 16 January 2003 Published online: 17 April 2003 Communicated by D. Buchholz and K.Fredenhagen     

Bicovariant differential calculus on quantum groupsSU q (N) andSO q (N)

Following Woronowicz's proposal the bicovariant differential calculus on the quantum groupsSU _q (N) andSO _q (N) is constructed. A systematic construction of bicovariant bimodules by using the matrix is presented. The relation between the Hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules, and theq-deformed universal enveloping algebras is given. Imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined. As an application the Maurer-Cartan equations and theq-analogue of the structure constants are formulated.Address after 1 Dec. 1990, Institute of Theoretical Physics, University of München.     

Some remarks on BRS transformations,anomalies and the cohomology of the Lie algebra of the group of gauge transformations

We show that ghosts in gauge theories can be interpreted as Maurer-Cartan forms in the infinite dimensional group ? of gauge transformations. We examine the cohomology of the Lie algebra of ? and identify the coboundary operator with the BRS operator. We describe the anomalous terms encountered in the renormalization of gauge theories (triangle anomalies) as elements of these cohomology groups.     

Graph reconstruction and quantum statistical mechanics

We study how far it is possible to reconstruct a graph from various Banach algebras associated to its universal covering, and extensions thereof to quantum statistical mechanical systems. It turns out that most the boundary operator algebras reconstruct only topological information, but the statistical mechanical point of view allows for complete reconstruction of multigraphs with minimal degree three.     

Symmetry groups and non-abelian cohomology

We consider the implementation of symmetry groups of automorphisms of an algebra of observables in a reducible representation whose multipliers in general are non-commuting operators in the commutant of the representation. The multipliers obey a non-abelian cocycle relation which generalizes the 2-cohomology of the group. Examples are given from the theory of spin algebras and continuous tensor products. For typeI representations we show that the multiplier can be chosen to lie in the centre, giving an isomorphism with abelian theory.Dedicated to Res Jost and Arthur Wightman